An Introduction to Dynamical Systems - 2nd Edition

Y7 >‘__

An Introduction to
Dynamical Systems

Continuous and Discrete,
Second Edition

R. Clark Robinson

_ _,_'»-~ '
<

» ‘->\‘_». _/



An Introduction to
Dynamical Systems

Continuous and Discrete,
Second Edition



Para
UNDERGRADUATE TEXTS - 19

An Introduction to
Dynamical Systems

Continuous and Discrete,
Second Edition

R. Clark Robinson

American Mathematical Society
Providence, Rhoda Island

EDITORIAL COMMITTEE

Paul J. Sally, Jr. (Chair) Joseph Silverman
Francis Su Susan Toiman

2010 Mathematics Subject Classi cation. Primary 34C}D(, 37Cx.x, 37Dxx,
37Exx, 37Nxx, 70K>cx.

This book was previously published by: Pearson Education, I.nc.

For additional information and updates on this book, visit
www.ams.org/bookpages/arnstexb19

Library of Congress Cataloging-in-Publication Data

Robinson, R. Clark (Rex Clark), 1943-
An introduction to dynamical systems 2 continuous and discrete / R. Clark Robinson. —Second

edition.
pages cm. -— (Pure and applied undergraduate texts ; volume 19)

Includes bibliographical references and index.
ISBN 978-0-B218-9135-3 (alk. paper)
1. Di erentiable dynamical systems. 2. Nonlinear theories. Chaotic behavior in systems. l. Title.

QA614.8.R.65 2012 2012025520
5l5'.39—dc23

Copying and reprinting. Individual readers of this publication, and nonpro t libraries
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief pa:-saga from this publication in
reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Acquisitions Department, American Mathematical Society,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
e-mail to repr1nt—perm:iss1on0ams . org.

© 2012 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page‘ at ht'tp://w\rw.ams.org/

10987654321 171615141312 i

How many are your works, O Lord!
In wisdom you made them all;

the earth is full of your creatures.
—Psalm 104-24



Contents

Preface xiii

Historical Prologue xvii

Part 1. Systems of Nonlinear Differential Equations 3

Chapter 1. Geometric Approach to Dilferential Equations 11
13
Chapter 2. Linear Systems 19
2.1. Fundamental Set of Solutions 21
Exercises 2.1 48
2.2. Constant Coe icients: Solutions and Phase Portraits 49
Exercises 2.2 52
2.3. Nonhomogeneous Systems: Time-dependent Forcing 52
Exercises 2.3 56
2.4. Applications 59
Exercises 2.4
2.5. Theory and Proofs 75
75
Chapter 3. The Flow: Solutions of Nonlinear Equations 83
3.1. Solutions of Nonlinear Equations 84
Exercises 3.1 96
3.2. Numerical Solutions of Differential Equations 97
Exercises 3.2
3.3. Theory and Proofs 109
109
Chapter 4. Phase Portraits with Emphasis on Fixed Points
4.1. Limit Sets _

vii

viii Contents

Exercises 4.1 114
4.2. Stability of Fixed Points 114
Exercises 4.2 119
4.3. Scalar Equations 119
Exercises 4.3 124
4.4. Two Dimensions and N ullclines 126
Exercises 4.4 133
4.5. Linearized Stability of Fixed Points 134
Exercises 4.5 143
4.6. Competitive Populations 145
Exercises 4.6 150
4.7. Applications 152
Exercises 4.7 158
4.8. Theory and Proofs 159

Chapter 5. Phase Portraits Using Scalar Functions 169
5.1. Predator—Prey Systems 169
Exercises 5.1 172
5.2. Undamped Forces 173
Exercises 5.2 182
5.3. Lyapunov Functions for Damped Systems 183
Exercises 5.3 190
5.4. Bounding Fhnctions 191
Exercises 5.4 195
5.5. Gradient Systems 195
Exercises 5.5 ‘ 199
5.6. Applications 199
Exercises 5.6 210
5.7. Theory and Proofs 210

Chapter 6. Periodic Orbits 213
6.1. Introduction to Periodic Orbits 214
Exercises 6.-1 218
6.2. Poincaré—Bend.ixs0n Theorem 219
Exercises 6.2 226
6.3. Self-Excited Oscillator 229
Exercises 6.3 232
6.4. Andronov—Hopf Bifurcation 232
Exercises 6.4 240
6.5. Homoclinic Bifurcation 242

Con tents

Exercises 6.5
6.6. Rate of Change of Volume
Exercises 6.6
6.7. Poincaré Map
Exercises 6.7
6.8. Applications
Exercises 6.8
6.9. Theory and Proofs

Chapter 7. Chaotic Attractors '
7.1. Attractors
Exercises 7.1
7.2. Chaotic Attractors
Exercise 7.2
7.3. Lorenz System
Exercises 7.3
7.4. Rbssler Attractor
Exercises 7.4
7.5. Forced Oscillator
Exercises 7.5
7.6. Lyapunov Exponents
Exercises 7.6
7.7. Test for Chaotic Attractors
Exercises 7.7
7.8. Applications
7.9. Theory and Proofs

Part 2. Iteration of Emotions

Chapter 8. Iteration of Flunctions as Dynamics
8.1. One-Dimensional Maps
8.2. Rmctions with Several Variables

Chapter 9. Periodic Points of One-Dimensional Maps
9.1. Periodic Points
Exercises 9.1
9.2. Iteration Using the Graph
Exercises 9.2
9.3. Stability of Periodic Points
Exercises 9.3
9.4. Critical Points and Basins

X Con tents

Exercises 9.4 390
9.5. Bifurcation of Periodic Points 391
Exercises 9.5 404
9.6. Conjugacy 406
Exercises 9.6 411
9.7. Applications 412
Exercises 9.7 416
9.8. Theory and Proofs 417

Chapter 10. Itineraries for One-Dimensional Maps 423
424
10.1. Periodic Points from Transition Graphs ' 435
437
Exercises 10.1 A 441
442
10.2. Topological ‘Ii-ansitivity 451
451
Exercises 10.2 454
455
10.3. Sequences of Symbols 463
464
Exercises 10.3 473
475
10.4. Sensitive Dependence on Initial Conditions 478
479
Exercises 10.4
487
10.5. Cantor Sets 487
490
Exercises 10.5 490
505
10.6. Piecewise Expanding Maps and Subshifts 507
513
Exercises 10.6 514
533
10.7. Applications 534
537
Exercises 10.7
541
10.8. Theory and Proofs 541

Chapter 11. Invariant Sets for One-Dimensional Maps
11.1. Limit Sets
Exercises 11.1
11.2. Chaotic Attractors
Exercises 11.2
11.3. Lyapunov Exponents
Exercises 11.3
11.4. Invariant Measures
Exercises 11.4 I
11.5. Applications
11.6. Theory and Proofs

Chapter 12. Periodic Points of Higher Dimensional Maps
12.1. Dynamics of Linear Maps

Contents

Exercises 12.1
12.2. Classi cation of Periodic Points
Exercises 12.2
12.3. Stable Manifolds
Exercises 12.3
12.4. Hyperbolic Toral Automorphisms
Exercises 12.4
12.5. Applications
Exercises 12.5
12.6. Theory and Proofs

Chapter 13. Invariant Sets for Higher Dimensional Maps

13.1. Geometric Horseshoe '

Exercises 13.1

13.2. Symbolic Dynamics

Exercises 13.2

13.3. Homoclinic Points and Horseshoes

Exercises 13.3

13.4. Attractors

Exercises 13.4

13.5. Lyapunov Exponents

Exercises 13.5

13.6. Applications .

13.7. Theory and Proofs

Chapter 14. Fractals

14.1. Box Dimension

Exercises 14.1

14.2. Dimension of Orbits

Exercises 14.2 ~

14.3. Iterated-Function Systems

Exercises 14.3

14.4. Theory and Proofs

Appendix A. Background and Terminology
A.1. Calculus Background and Notation
A.2. Analysis and Topolog Terminology
A.3. Matrix Algebra

Appendix B. Generic Properties

Bibliography

Index



Preface

Preface to the Second Edition

In the second edition of this book, much of the material has been rewritten to clarify
the presentation. It also has provided the opportunity for correcting many minor
typographical errors or mistakes. Also, the de nition of a chaotic attractor has been
changed to include the requirement that the chaotic attractor is transitive. This
is the usual de nition and it eliminates some attractors that should not be called
chaotic. Several new applications are included for systems of differential equations
in Part 1. I would encourage readers to email me with suggestions and further
corrections that are needed.

, R. Clark Robinson
March 2012

Preface to the First Edition

This book is intended for an advanced undergraduate course in dynamical sys-
tems or nonlinear ordinary di erential equations. There are portions that could
be bene cially used for introductory master level courses. The goal is a treatment
that gives examples and methods of calculation, at the same time introducing the
mathematical concepts involved. Depending on the selection of material covered,
an instructor could teach a course from this book that is either strictly an intro-
duction into the concepts, that covers both the concepts on applications, or that
is a more theoretically mathematical introduction to dynamical systems. Further
elaboration of the variety of uses is presented in the subsequent discussion of the
organization of the book.

=The assumption is that the student has taken courses on calculus covering both
single variable and multivariables, a course on linear algebra, and an introductory

i
xiii

xiv Preface

course on differential equations. From the multivariable calculus, the material on
partial derivatives is used extensively, a.nd in a few places multiple integrals and sur-
face integrals are used. (See Appendix A.1.) Eigenvalues and eigenvectors are the
main concepts used from linear algebra, but further topics are listed in Appendix
A.3. The material from the standard introductory course on differential equations
is used only in Part 1; we assume that students can solve rsborder equations by
separation of variables, and that they know the form of solutions from second-order
scalar equations. Students who have talcen an introductory course on differential
equations are usually familiar with linear systems with constant coef cients (at
least the real-eigenvalue case), but this material is repeated in Chapter 2, where we
alsdlintroduce the reader to the phase portrait. At Northwestern, some students
have taken the course covering -part one on differer'1_tial equations without this in-
troductory course on differential equations; they have been able to understand the
new material when they have been willing to do the extra work in a few areas that
is required to ll in the missing background. Finally, we have not assumed that
the student has had a course on real analysis or advanced calculus. However, it
is convenient to use some of the terminology from such a course, so we include an
appendix with terminology on continuity and topology. (See Appendix A.)

Organization

This book presents an introduction to the concepts of dynamical systems. It
is divided into two parts, which can be treated in either order: The rst part
treats various aspects of'systems of nonlinear ordinary differential equations, and
the second part treats those aspects dealing with iteration of a function. Each
separate part can be used for a one-quarter course, a one-semester course, a two-
quarter com'se, or possibly even a year course. At Northwestern University, we have
courses that spend one quarter on the rst part and two quarters on the second
part. In a one-quarter course on differential equations, it is di icult to cover the
material on chaotic attractors, even skipping many of the applications and proofs
at the end of the chapters. A one-semester course on differential equations could
also cover selected topics on iteration of functions from Chapters 9-11. In the
course on discrete dynamical systems using Part 2, we cover most of the material
on iteration of one-dimensional functions (Chapters 9-11) in one quarter. The
material on iteration of functions in higher dimensions (Chapters 12—13) certainly
depends on the one-dimensional material, but a one-semester course could mix
in some of the higher dimensional examples with the treatment of Chapters 9-
11. Finally, Chapter 14 on fractals could be treated after Chapter 12. Fractal
dimensions could be integrated into the material on chaotic attractors at the end of
a course on differential equations. The material on fractal dimensions or iterative
function systems could be treated with a course on iteration of one-dimensional
flll'lCl;l0l1S.

The main concepts are presented in the rst sections of each chapter. These
sections are followed by a section that presents some applications and then by
a section that contains proofs of the more difficult results and more theoretical
material. The division of material between these types of sections is somewhat
arbitrary. The theorems proved at the end of the chapter are restated with __their

Preface xv

original theorem number. The material on competitive populations and predator—
prey systems is contained in one of the beginning sections of the chapters in which
these topics are covered, rather than in the applications at the end of the chapters,
because these topics serve to develop the main techniques presented. Also, some
proofs are contained in the main sections when they are more computational and
serve to make the concepts clearer. Longer and more technical proofs and further
theoretical discussion are presented separately at the end of the chapter.

A course that covers the material from the primary sections, without covering
the sections at the end of the chapter on applications and more theoretical material,
results in a course on the concepts of dynamical systems with some motivation from
applications.

The applications provide motivation and illustrate the usefulness of the con-
cepts. None of the material from ‘the sections on applications is necessary for
treating the main sections of later chapters. Treating more of this material would
result in a more applied emphasis.

Separating the harder proofs allows the instructor to determine the level of
theory of the c01u'se taught using this book as the text. A more theoretic course
could consider most of the proofs at the end of the chapters.

Computer Prograrns

This book does not explicitly cover aspects of computer programming. How-
ever, a few selected problems require computer simulations to produce phase por-
traits of differential equations or to iterate functions. Sample Maple worksheets,
which the students can modify to help with some of the more computational prob-
lems, will be available on the webpage: '

http://www.math.northwestern.edu/~clark/dyn-sys.

(Other material on corrections and updates of the book will also be available at
this website.) There are several books available that treat dynamical systems in
the context of Maple or Mathematica: two such books are [Kul02] by M. Kulen-
ovié and [Lyn01] by S. Lynch. The book [P0l04] by J. Polking and D. Arnold
discusses using Matlab to‘ solve differential equations using packages available at
http://math.rice.edu/~d eld. The book [Nus98] by H. Nusse and J. Yorke
comes with its own specialized dynamical systems package.

Acknowledgments

I would like to acknowledge some of the other books I have used to teach this
material, since they have in uenced my understanding of the material, especially
with regard to effective ways to present material. I will not attempt to list more
advanced books which have also affected my understanding. For the material on
differential equations, I have used the following books: F. Brauer and J. Nohel
[Bra69], M. Hirsch and S. Smale [Hir74], M. Braun [Bra73], I. Percival and
D. Richards [Per82], D.W. Jordan and P. Smith [Jor87], J. Hale and H. Kocak
[Ha.l91], and S. Strogatz [Str94]. For the material on iteration of functions, I have
used the following books: the two books by R. Devaney [Dev89] and [Dev92],
D. Gulick [Gul92], and K. Alligood, T. Sauer, and J. Yorke [A1197].

xvi Preface

I would also like to thank three professors under whom I studied while a grad-
uate student: Charles Pugh, Morris Hirsch, and Stephen Smale. These people
introduced me to the subject of dynamical systems and taught me many of the
ideas and methods that I have used throughout my career. Many of my colleagues
at Northwestern have also deeply in uenced me in different ways: these people
include John Franks, Donald Saari, and Robert Williams.

I thank the following reviewers for their comments and useful suggestions for
improvement of the manuscript:

John Alongi, Pomona College
Pau Atela, Smith College
L Peter Bates, Brigham Young University
Philip Bayly, Washington University
Michael Brin, University of Maryland
Roman Grigoriev, Georgia Technological Institute
Palle Jorgensen, University of Iowa
Randall Pyke, Ryerson University
Joel Robbin, University of Wisconsin
Bjorn Sandstede, Ohio State University
Douglas Shafer, University of North Carolina at Charlotte
Milena Stanislavova, University of Kansas
Franz Tanner, Michigan Technological University
Howard Weiss, Pennsylvania State University

I also thank Miguel Lerma for help in solving various IMEX and graphics prob-
lems, Marian Gidea for help with Adobe Illustrator, and Kamlesh Parwani for help
with some Maple worksheets.

I gratefully acknowledge the photograph by Julio Ottino and Paul Swanson
used on the cover of the rst edition of the book depicting mixing of uids. This
photo had previously appeared in the article [Ott89b]. A brief discussion of his
research is given in Section 11.5.4.

Most of all, I am grateful to my wife Peggie, who endured, with patience,
miderstanding, and prayer, the ups and downs of writing this book from inception
to conclusion.

R. Clark Robinson
[email protected]
October 2003

Historical Prologue

The theory of differential equations has a long history, beginning with Isaac Newton.
From the early Greeks through Copernicus, Kepler, and Galileo, the motions of
planets had been described directly in terms of their properties or characteristics,
for example, that they moved on approximately elliptical paths (or in combinations
of circular motions of different periods and amplitudes). Instead of this approach,
Newton described the laws that determine the motion in terms of the forces acting
on the planets. The effect of these forces can be expressed by differential equations.
The basic law he discovered was that the motion is determined by the gravitational
attraction between the bodies, which is proportional to the product of the two
masses of the bodies and one over the square of the distance between the bodies.
The motion of one planet around a sun obeying these laws can then be shown to lie
on an ellipse. The attraction of the other planets could then explain the deviation
of the motion of the planet from the elliptic orbit. This program was continued
by Euler, Lagrange, Laplace, Legendre, Poisson, Hamilton, Jacobi, Liouville, and
others.

By the end of the nineteenth century, researchers realized that many nonlinear
equations did not have explicit solutions. Even the case of three masses moving
under the laws of Newtonian attraction could exhibit very complicated behavior
and an explicit solution was not possible (e.g., the motion of the smi, earth, and
moon cannot be given explicitly in terms of known fimctions). Short term solutions
could be given by power series, but these were not useful in determining long-tenn
behavior. Poincaré, working from 1880 to 1910, shifted the focus from nding
explicit solutions to discovering geometric properties of solutions. He introduced
many of the ideas in speci c examples, which we now group together under the
heading of chaotic dynamical systems. In particular, he realized that a deterministic
system (in which the outside forces are not varying and are not random) can exhibit
behavior that is apparently random (i.e., it is chaotic).

In 1898, Hadamard produced a speci c example of geodesics on a surface of
constant negative curvature which had this property of chaos. G. D. Birkho

_

xvii

xviii Historical Prologue

continued the work of Poincaré and found many different types of long-term limiting
behavior, including the a- and w-limit sets introduced in Sections 4.1 and 11.1. His
work resulted in the book [Bir27] from which the term “dynamical systems” comes.

During the rst half of the twentieth century, much work was carried out on
nonlinear oscillators, that is, equations modeling a collection of springs (or other
physical forces such as electrical forces) for which the restoring force depends non-
linearly on the displacement from equilibriurn. The stability of xed points was
studied by several people including Lyapunov. (See Sections 4.5 and 5.3.) The ex-
istence of a periodic orbit for certain self-excited systems was discovered by Van der
Pol. (See Section 6.3.) Andronov and Pontryagin showed that a system of di. 'eren-
tial equations was structurally stable near an attracting xed point, [And37] (i.e.,
the solutions for a small perturbation of the differential equation could be matched
with the solutions for the original equations). Other people carried out research
on nonlinear differential equations, including Bendixson, Cartwright, Bogoliubov,
Krylov, Littlewood, Levinson, and Lefschetz. The types of solutions that could be
analyzed were the ones which settled down to either (1) an equilibrium state (no
motion), (2) periodic motion (such as the rst approximations of the motion of the
planets), or (3) quasiperiodic solutions which are combinations of several periodic
terms with incommensurate frequencies. See Section 2.2.4. By 1950, Cartwright,
Littlewood, and Levinson showed that a certain forced nonlinear oscillator had in-

nitely many different periods; that is, there were in nitely many different initial
conditions for the same system of equations, each of which resulted in periodic mo-
tion in which the period was a multiple of the forcing frequency, but different initial
conditions had different periods. This example contained a type of complexity not
previously seen.

In the 1960s, Stephen Smale returned to using the topological and geometric
perspective initiated by Poincaré to understand the properties of differential equa-
tions. He wrote a very in uential survey article [Sma67] in 1967. In particular,
Smale’s “horseshoe” put the results of Cartwright, Littlewood, and Levinson in a
general framework and extended their results to show that they were what was later
called chaotic. A group of mathematicians worked in the United States and Europe
to flesh out his ideas. At the same time, there was a group of mathematicians in
Moscow lead by Anosov and Sinai investigating similar ideas. (Anosov generalized
the work of Hadamard to geodesics on negatively curved manifolds with variable
curvature.) The word “chaos” itself was introduced by T.Y. Li and J. Yorke in
1975 to designate systems that have aperiodic behavior more complicated than
equilibrium, periodic, or quasiperiodic motion. (See [Li,75].) A related concept
introduced by Ruelle and Takens was a strange attractor. It emphasized more the
complicated geometry or topology of the attractor in phase space, than the com-
plicated nature of the motion itself. See [Rue71]. The theoretical work by these
mathematicians supplied many of the ideas and approaches that were later used
in more applied situations in physics, celestial mechanics, chemistry, biology, and
other elds.

Historical Prologue xix

The application of these ideas to physical systems really never stopped. One
of these applications, which has been studied since earliest times, is the descrip-
tion and determination of the motion of the planets and stars. The study of the
mathematical model for such motion is called celestial mechanics, and involves a

nite number of bodies moving under the effects of gravitational attraction given
by the Newtonian laws. Birkhoff, Siegel, Kolmogorov, Arnold, Moser, Herman,
and many others investigated the ideas of stability and found complicated behavior
for systems arising in celestial mechanics and other such physical systems, which
could be described by what are called Hamiltonian di erential equations. (These
equations preserve energy and can be expressed in terms of partial derivatives of
the energy function.) K. Sitnikov in [Sit60] introduced a situation in which three
masses interacting by Newtonian attraction can exhibit chaotic oscillations. Later,
Alekseev showed that this could be understood in terms of a “Smale horseshoe",
[Ale68a], [Ale68b], and [Ale69]. The book by Moser, [Mos73], made this result
available to many researchers and did much to further the applications of horse-
shoes to other physical situations. In the 1971 paper [Rue71] introducing strange
attractors, Ruelle and Takens indicated how the ideas in nonlinear dynamics could
be used to explain how turbulence developed in uid ow. Further connections
were made to physics, including the periodic doubling route to chaos discovered by
Feigenbaum, [Fei78], and independently by P. Coullet and C. 'I\'esser, [Cou78].

Relating to a completely dilferent physical situation, starting with the work of
Belousov and Zhabotinsky in the 1950s, certain mathematical models of chemical
reactions that exhibit chaotic behavior were discovered. They discovered some
systems of differential equations that not only did not tend to an equilibrium, but
also did not even exhibit predictable oscillations. Eventually, this bizarre situation
was understood in terms of chaos and strange attractors.

In the early 1920s, A.J. Lotka and V. Volterra independently showed how dif-
ferential equations could be used to model the interaction of two populations of
species, [Lot25] and [V0131]. In the early 1970s, May showed how chaotic out-
comes could arise in population dynamics. In the monograph [May75], he showed
how simple nonlinear models could provide “mathematical metaphors for broad
classes of phenomena." Starting in the 1970s, applications of nonlinear dynamics
to mathematical models in biology have become widespread. The undergraduate
books by Murray [Mur8§] and Taubes [Tau01] afford good introductions to bio-
logical situations in which both oscillatory and chaotic differential equations arise.
The books by Kaplan and Glass [Kap95] and Strogatz [Str94] include a large
number of other applications.

Another phenomenon that has had a great impact on the study of nonlinear
differential equations is the use of computers to find numerical solutions. There
has certainly been much work done on deriving the most efficient algorithms for
carrying out this study. Although we do discuss some of the simplest of these,
our focus is more on the use of computer simulations to nd the properties of
solutions. E. Lorenz made an important contribution in 1963 when he used a
computer to study nonlinear equations motivated by the turbulence of motion of
the atmosphere. He discovered that a small change in initial conditions leads to
very different outcomes in a relatively short time; this property is called sensitive

xx Historical Prologue

dependence on initial conditions or, in more common language, the butter y effect.
Lorenz used the latter term because he interpreted the phenomenon to mean that
a butter y apping its wings in Australia today could affect the weather in the
United States a month later. We describe more of his work in Chapter 7. It
was not until the 1970s that Lorenz’s work became known to the more theoretical
mathematical community. Since that time, much effort has gone into showing
that Lorenz’s basic ideas about these equations were correct. Recently, Warwick
Tucker has shown, using a computer-assisted proof, that this system not only has
sensitive dependence on initial conditions, but also has what is called a “chaotic
attractor”. (See Chapter 7.) About the same time as Lorenz, Ueda discovered that
a periodically forced Van der Pol system (or other nonlinear oscillator) has what is
now called a chaotic attractor. Systems of this type are also discussed in Chapter
7. (For a later publication by Ueda, see also [Ued9_2].)

Starting about 1970 and still continuing, there have been many other numer-
ical studies of nonlinear equations using computers. Some of these studies were
introduced as simple examples of certain phenomena. (See the discussion of the
Rossler Attractor given in Section 7.4.) Others were models for speci c situations
in science, engineering, or other elds in which nonlinear differential equations are
used for modeling. The book [Enn97] by Enns and McGuire presents many com-
puter programs for investigation of nonlinear mctions and differential equations
that arise in physics and other scienti c disciplines.

In sum, the last 40 years of the twentieth century saw the growing importance
of nonlinearity in describing physical situations. Many of the ideas initiated by
Poincaré a century ago are now much better understood in terms of the mathematics
involved and the way in which they can be applied. One of the main contributions
of the modern theory of dynamical systems to these applied elds has been the
idea that erratic and complicated behavior can result from simple situations. Just
because the outcome is chaotic, the basic environment does not need to contain
stochastic or random perturbations. The simple forces themselves can cause chaotic
outcomes.

There are three books of a nontechnical nature that discuss the history of
the development of “chaos theory”: the best seller Chaos: Making a New Science
by James Gleick [Gle87], Does God Play Dice?, The Mathematics of Chaos by
Ian Stewart [Ste89], and Celestial Encounters by Florin Diacu and Philip Holmes
[Dia96]. Stewart’s book puts a greater emphasis on the role of mathematicians in
the development of the subject, while Gleick’s book stresses the work of researchers
making the connections with applications. Thus, the perspective of Stewart’s book
is closer to the one of this book, but Gleick’s book is accessible to a broader audience
and is more popular. The book by Diacu and Holmes has a good treatment of
Poinca.ré’s contribution and the developments in celestial mechanics up to today.

Part 1

Systems of Nonlinear
Differential Equations



i
Chapter 1

Geometric Approach to
Differential Equations

In a basic elementary differential equations coLn'se, the emphasis is on linear dif-
ferential equations. One example treated is the linear second-order differential
equation
(1.1) m:i':+b:i:+k:c=O,
where we wri.te 1.: for -rlJw; and :1..: for Wdzzr, and m,Ic > 0 and b 2 0. Thi.s equati.on
is a model for a linear spring with friction and is also called a damped harmonic
oscillator.

To determine the future motion, we need to know both the cLn'rent position
:c and the current velocity a':. Since these two quantities determine the motion,
it is natural to use them for the coordinates; writing v = a':, we can rewrite this
equation as a linear system of rst-order differential equations (involving only first
derivatives), so mi: = mi = —I<:a: — bv:
(1.2) J3 = v,

v_ = — —k :r: — —b v.
mm

In matrix notation, this becomes
- O1

('f0) = (_ _k _ _b) v

mm
For a linear differential equation such as (1.1), the usual solution method is to
seek solutions of the form :c(t) = e“, for which :i: = ,\ e’\‘ and it = /\2 e’\‘. We
need

mA2e’\‘+b/\e’\‘+l<:e'\‘=0,

3

4 1. Geometric Approach to Differential Equations

or

(1.3) m,\’+b,\+k=o,

which is called the characteristic equation.

When b = 0, the solution of the characteristic equation is /\2 = _k!m or

/\ = zlziw, where w = Since e"“" = cos(wt) + i sin(w t), the real and

imaginary parts, cos(w t) and sin(w t), are each solutions. Linear combinations are

also solutions, so the general solution is

:z(t) = A cos(wt) + B sin(wt),

M v(t) = —wA sin(wt) + wB cos(wt),

where A and B are arbitrary constants. These solutions are both periodic, with

the same period 21r/w. See Figure 1. -

4I

2

t
OI

Z1!

cu

Figure 1. Solutions for the linear harmonic oscillator: a: as a function of t

Another way to understand the solutions when b = 0 is to nd the energy that
is preserved by this system. Multiplying the equation 1) + wz :1: = 0 by v = :i:, we
get

'u 1) + wz ma’: = 0.
The left-hand side of this equation is the derivative with respect to t of

1 wz
E(:n, v) = 5122 + ?:r2,

so lls function E(a:, v) is a constant along a solution of the equation. This integral
of motion shows clearly that the solutions move on ellipses in the (a:, v)-plane given
by level sets of E. There is a xed point or equilibrium at (:1:, 'u) = (0,0) and other
solutions travel on periodic orbits in the shape of ellipses around the origin. For this
linear equation, all the orbits have the same shape and period which is independent
of size: We say that the local and global behavior is the same. See Figure 2.

For future reference, a point x‘ is called a xed point of a system of di 'erential
equations it = F(x) provided F(x") = 0. The solution starting at a xed point has
zero velocity, so it just stays there. Therefore, if x(t) is a solution with x(0) = x"',
then x(t) = X‘ for all t. Traditionally, such a point was called an equilibr-ium point
because the forces are in balance and the point does not move.

1. Geometric Approach to Differential Equations 5

A point x‘ is called periodic for a system of differential equations 5: = F(x),
provided that there is some T > 0 such that the solution x(t) with initial condition
x(0) = x‘ has x(T) = x‘ but x(t) 79 x‘ for 0 < t < T. This value T is called the
period or least period. It follows that x(t + T) = x(t) (i.e., it repeats itself after T
units of time). The set of points {x(t) : O 5 t 3 T} is called the periodic orbit.

Figure 2. Solutions for the linear harmonic oscillator in (z, u)-space.

The set of curves in the (rt, v)-plane determined by the solutions of the system
of differential equations is an example of a phase portrait, which we use throughout
this book. These portraits are a graphical (or geometric) way of understanding the
solutions of the differential equations. Especially for nonlinear equations, for which
we are often unable to obtain analytic solutions, the use of graphical representations
of solutions is important. Besides determining the phase portrait for nonlinear
equations by an energy function such as the preceding, we sometimes use geometric
ideas such as the nullclines introduced in Section 4.4. Other times, we use numerical
methods to draw the phase portraits.

Next, we consider the case for b > 0. The solutions of the characteristic equa-
tion (1.3) is

A = T-b 5: \/bi — 4km = —c;l: Z. ].L,

where

c=% and p=i/%—c2.

(These roots are also the eigenvalues of the matrix of the system of differential
equations.) The general solution of the system of equations is

a:(t) = e""‘ [A cos(pt) + B sin(p.t)],

v(¢) = 5°‘ [— (An + BC) 8in(/M) + (Bil — Ac) <=<>$(#¢)l ,

where A and B are arbitrary constants.

Another way to understand the properties of the solutions in this case is to use

the “energy function”, 2

E(:z:,'u) = £112 + %:|:2,

6 1. Geometric Approach to Differential Equations

for w = \/ E/m, which is preserved when b = 0. For the system with b > 0 and so
c > 0,

&d E(a:,v)-v11. +w2 22:1.:

= v(—2cv — wzm) + wzarv

= -2w’ 50.

This shows that the energy is nonincreasing. A simple argument, which we give
in the section on Lyapimov functions, shows that all of the solutions go to the
xed point at the origin. The use of this real-valued function E(a:,v) is a way to
show that the origin is attracting without using the explicit representation of the
solutions.

The system of equations (1.2) are linear, and most of the equations we consider
are nonlinear. A simple nonlinear example is given by the pendulum

mL§ = —mg sin(0).
Setting :c = 0 and '0 = 9, we get the system
(1.4) :i: = v,

ii = —% sin(a:).
It is difficult to get explicit solutions for this nonlinear system of equations. How-
ever, the “energy method” just discussed can be used to nd important properties
of the solutions. By a derivation similar to the foregoing, we see that

2
-E(:c,'u) = U? +1— %cos(a:)
is constant along solutions. Thus, as in the linear case, the solution moves on a
level set of E, so the level sets determine the path of the solution. The level sets
are given in Figure 3 without justi cation, but we shall return to this example in
Section 5.2 and give more details. See Example 5.3.

‘U

\/ Y

Figure 3. Level sets of energy for the pendulum

There are xed points at (a:,11) = (0,0), (:l:1r,0). The solutions near the orign
are periodic, but those farther away have :1: either monotonically increasing or

1. Geometric Approach to Di erential Equations 7

decreasing. We can use the level curves and trajectories in the phase space (the
(az, v)-space) to get information about the solutions.

Even this simple nonlinear equation illustrates some of the differences between
linear and nonlinear equations. For the linear harmonic oscillator with b = 0, the
local behavior near the xed point determines the behavior for all scales; for the
pendulum, there are periodic orbits near the origin and nonperiodic orbits farther
away. Second, for a linear system with periodic orbits, all the periods are the same;
the period varies for the pendulum. The plots of the three solutions which are time
periodic are given in Figure 4. Notice that the period changes with the amplitude.
Finally, in Section 2.2 we give an algorithm for solving systems of linear differential
equations with constant coe icients. On the other hand, there is no simple way to
obtain the solutions of the pendulum equation; the energy method gives geometric
information about the solutions, but not explicit solutions.

41¢

2»

t

0.

16

_g -

-4]

Figure 4. Time plots of solutions with different amplitudes for the pendulum
showing the variation of the period

Besides linear systems and the pendulum equation, we consider a few other
types of systems of nonlinear differential equation including two species of popu-
lations, which either compete or are in a predator—prey relationship. See Sections
4.6 and 5.1. Finally, the Van der Pol oscillator has a unique periodic orbit with
other solutions converging to this periodic orbit. See Figure 5 and Section 6.3.
Basically, nonlinear differential equations with two variables cannot be much more
complicated than these examples. See the discussion of the Poinca.ré—Bendixson
Theorem 6.1.

In three and more dimensions, there can be more complicated systems with
apparently “chaotic” behavior. Such motion is neither periodic nor quasiperiodic,
appears to be random, but is determined by explicit nonlinear equations. An
example is the Lorenz system of differential equations

ri: = —10:r+ 10y,
(1.5) 1] = 28a:—y—a:z,

2—_ —3§z+1:y.

8 1. Geometric Approach to Differential Equations

I2

$1

Figure 5. Phase portrait of the Van der Pol oscillator

Trajectories originating from nearby initial conditions move apart. Such systems
are said to have sensitive dependence on initial conditions. See Figure 6 where two
solutions are plotted, each starting near the origin. As these solutions are followed
for a longer time, they ll up a set that is a fuzzy surface within three dimensions.
(The set is like a whole set of sheets stacked very close together.) Even though the
equations are deterministic, the outcome is apparently random or chaotic. This
type of equation is discussed in Chapter 7. The possibility of chaos for nonlinear
equations is an even more fundamental difference from linear equations.

_ ~ ’I _

\\‘s\‘i"’J///(./1. ” O

&J/\ / l I

Figure 6. Two views of the Lorenz attractor
Finally, we highlight the differences that can occm" for time plots of solutions
of di erential equations. Figure 7(a) shows an orbit that tends to a constant value
(i.e., to a. xed point). Figure 7(b) shows a periodic orbit that repeats itself after the
time increases by a xed amount. Figure 7(0) has what is called a quasiperiodic
orbit; it is generated by adding together functions of two di erent periods so it
never exactly repeats. (See Section 2.2.4 for more details.) Figure 7(d) contains a
chaotic orbit; Section 7.2 contains a precise de .nition, but notice that there is no
obvious regularity to the length of time of the different types of oscillation.

1. Geometric Approach to Differential Equations z
$1 $1

lz

_ (a) (b)

. 11 I1

.2i1i |1 .l,1|, I,I] I1,.|1; Ii l iia| l .1 i

(C) (<1)

Figure 7. Plots of one position variable as a function of t: (a) orbits tending
to a xed point, (b) a periodic orbit, (c) a quasiperiodic orbit, and (d) a chaotic
orbit.



i
Chapter 2

Linear Systems

This book is oriented toward the study of nonlinear systems of differential equa-
tions. However, aspects of the theory of linear systems are needed at various places
to analyze nonlinear equations. Therefore, this chapter begins our study by present-
ing the solution method and properties for a general n-dimensional linear system
of differential equations. We quickly present some of the material considered in a
beginning differential equation course for systems with two variables while general-
izing the situation to n variables and introducing the phase portrait and other new
aspects of the qualitative approach.

2
Remember that throughout the book we write :1": for % and for

In Chapter 1 we mentioned that a'linear second-order differential equation of
the form

m:'i'+ka:+b:i:=0

can be written as a linear system of rst-order di erential equations; setting 2:; = a:
and 2:2 = :i:,

ii = $2.

w. 2=——kw1——b=vz,
mm

or in matrix notation

@

X = < _k; __ E‘
§°>—1 L)

where x = (I.11: ) 1. 5 a vector.
2

Z
11

12 2. Linear Systems

In this chapter, we study general systems of linear differential equations of this
type with n variables with constant coefficients,

ii = an :v1 + 01,2 $2 + - -- + =11... 311)
J32 = ¢12,1$1+ 112,2 $2 + - - ' + 0-2,11 In.

in =an,1171 +v»..,zIz+---+¢1..,-.1...
where all the a,-__,- are constant real numbers. This system can be rewritten using
vector and matrix notation. Let A be the n X n matrix with given constant real
entries a.,-__,-, and x is the (column) vector in IR" of the variables,

.121
x=(a:1,...,:z:,,)T= E .

Tn

Here, ($1, . . . ,a:,,)T is the transpose of the row vector which yields a column vector.
The system of linear differential equations can be written as

(2.1) >z= Ax.

At times, we study linear systems that depend on time, so the coe icients
a,,_.,- (t) and entries in the matrix A(t) are functions of time t. This results in a
time-dependent linear system

(2.2) it = A(t) x.

This type of equation could arise if the spring constants could somehow be controlled
externally to give an equation of the form

:'t+k(t):r=0,

where k(t) is a known function of t. In this book, these equations mainly occur
by “linearizing” along a solution of a nonlinear equation. See the First Variation
Equation, Theorem 3.4.

As discussed in Chapter 1, equation (1.1) has the two linearly independent
solutions e'°‘cos(at) and e"°‘sin(/i t), where c = b/2-m. and #2 = '°/m — c2. All
solutions are linear combinations of these two solutions. In Section 2.1, we show
that for linear systems with n-variables, we need to nd n independent solutions.
We also check that we can take linear combinations of these solutions and get all
other solutions, so they form a “fundamental set of solutions." In Section 2.2, we
see how to nd explicit solutions for linear systems in which the coefficients do not
depend on time, i.e., linear systems with constant coe icients. We also introduce the
phase portrait of a system of differential equations. Section 2.2.4 contains examples
with four variables in which the solutions have sines and cosines with two different
frequencies and not just one, i.e., the solution is quasiperiodic. The motion of such a
quasiperiodic solution is complicated but does not appear random. Therefore, when
we de ne chaotic sets for nonlinear systems in Section 7.2, we require that there is
an orbit that winds throughout the set and is neither periodic nor quasiperiodic.
Finally, Section 2.3 considers linear equations in which there is a “forcing vector”
besides the linear term. Nonhomogeneous systems are used several times later in

2.1. Fundamental Set of Solutions 13

the book: (1) to show that the linear terms dominate the higher order terms near
a xed point (Section 4.8), (2) to give an example of a Poincaré map (Section 6.7),
and (3) to give an example of a calculation of a Lyapunov exponent (Section 7.6).

2.1. Fundamental Set of Solutions

In the next section, we give methods of nding solutions for linear equations with
constant coefficients. This section focuses on showing that we need n independent
solutions for a system with n variables. We also show that a linear combination
of solutions is again a solution. Finally, we show how the exponential of a matrix
can be used to nd an analytical, but not very computable, solution of any system
with constant coefficients. This exponential is used in the next section to motivate
the method of nding solutions of systems with repeated eigenvalues.

We give the de nitions for the time-dependent case, but there the de nitions
of the time-independent case given by equation (2.1) are essentially the same.
Solution

A solution of equation (2.2) is a function x(t) of time on some open interval of
time where A(t) is de ned such that

ditxe) = A(t) x(t).

A solution satis es the initial condition X0, provided x(0) = xo.

Existence of solutions ,

In Section 2.2, we construct solutions for constant coef cient systems of linear
equations. Together with the discussion of fundamental matrix solutions, this dis-
cussion shows that, given an initial condition xo, there is a solution x(t) satisfying
the differential equation with x(0) = xo. For time-dependent linear systems, there
is no such constructive proof of existence; however, the result certainly follows from
Theorem 3.2 for nonlinear systems of differential equations. There are also easier
proofs of existence that apply to just such time-dependent linear systems, but we
do not discuss them.

Uniqueness of solutions

The uniqueness of solutions follows from Theorem 3.2 for systems of nonlinear
differential equations. However, we can give a much more elementary proof in this
case, so we state the result here and give its proof in Section 2.5.

Theorem 2.1. Let A(t) be an n x n matria: whose entries depend continuously
on t. Given xo in 1R, there is at most one solution x(t) of x = A(t)x with
x(0) = xo.

Linear combinations of solutions

For j = 1, . . . , lc, assume that x5(t) are solutions of equation (2.2) (de ned on
a common time interval) and Cj are real or complex scalars. Using the properties

14 2. Linear Systems

of differentiation and matrix multiplication,

-dE(¢,x1(t) + ~ - - + ck x'°(t)) = ¢,>'¢‘(t) + - -- +¢r>'<’°(i)

= e, A(t)x1(t) + +ck A(t) '°(t)

= A(t) (c1x1(t) + - -- + ckxk( 5 y/X ,

so the linear combination cl x1(t) + - - - + ck x" (t) is also a solution. Thus, a linear
combination of solutions is a solution.
Matrix solutions

To explain why it is enough to nd n solutions for a linear system with n
variables, we introduce matrix solutions and then the Wronskian. If x-’(t) for
1 5 j 3 n are n-solutions of equations (2.1) or (2,2), we can combine them into a
single matrix by putting each solution as a column,

M(t) = (X10), - - - ,X"(¢))-

By properties of differentiation and matrix multiplication,

%M(z) = (>'<‘(t), . . . ,x"(i))
= (A(t) x1(t), . . . , A(t) x"(t))
= A(t) (xl (t), . . . ,x"(t))

= A(t) M(t).

Thus, M(t) is a matrix that satis es the differential equation.
Any n x k matrix M((t) that satis es

%M@=A@M@

is called a. matrix solution. If M(t) is a matrix solution with columns {x7(t)}_f=1
and we write c for the constant vector with entries cj, then

%M®c=MQM®q

and so M(t)c = cl x1(t) + - - - + ck x"(t) is the vector solution that satis es

M(0)c = c1 x1(0) + - -- + ck xk(0)

when t = 0. In particular, if we let c = u-7 be the vector with a 1 in the j ‘
coordinate and 0 in the other coordinates, then we see that M(t)u-" = x1 (t), which
is the j"‘ column of M(t), is a vector solution. The next theorem summarizes the
preceding discussion.

Theorem 2.2. Assume that xj (t) are solutions of equation (2.2) for j = 1, . . . , lc.

a. If cj are real or complea: scalars forj = 1, . . . ,k, then 25:, c_,- x7(t) is a

(vector) solution. '

b. The matria: M(t) = (x1(t), . . . ,x'°(t)) is a matria: solution.

c. If c = (c1, . . . , c;,)T is a constant vector and M(t) is an nxk matria: solution,
then M(t)c is a (vector) solution.

2.1. Fundamental Set of Solutions 15

d. If M(t) -is a matria: solution of equation (2.2), then the columns of M(t) are
vector solutions.

Fundamental set of solutions
A set of vectors {vi };=1 is linearly independent provided that, whenever a

linear combination of these vectors gives the zero vector,

c1v1+---+c,,v"=0,

then all the cj = 0. If {v-'l };=1 is a set of n vectors in IR", then they are linea.rly
independent if and only if

det (v1,...,v") 750.

If {V7 };‘=l is a set of n linearly independent vectors in IR", then for any x° in IR"
there exist yl, ..., yn such that

x° = y1v1+---+y,,v".

For this reason, a li.nea.rly independent set of vectors {v-"' };'=l is called a basis of
R".

A set of solutions {x1(t),...,x"(t)} is linearly independent provided that,
whenever

c1x1 (t) + - - - + c,,x"(t) = 0 for all t,

then all the cj = O. When a set of n solutions {x1(t),. ..,x"(t)} is linearly in-
dependent, then it is called a fundamental set of solutions, and the corresponding
matrix solution M(t) is called a fundamental matrix: solution.

Just as for vectors, we can relate the condition of being linearly independent
to a determinant. If M(t) is an n x n matrix solution of equations (2.1) or (2.2),
then the determinant

W(t) = det (M(t))

is called the Wronskian of the system of vector solutions given by the columns of
M(t).

If M(t) is an n X n matrix solution with W(0) = det(M(0)) 96 O, then for any
xo, we can solve the equation M(O)c = xo for c to give a solution x(t) = M(t)c
with x(0) = xo. Notice that, by linear algebra, we need at least n columns in order
to be able to solve M(O)c = xo for all initial conditions xo.

The Liouville formula, (2.3), given in the next theorem, relates W(t) and W(t0)
at different times. This formula implies that, if W(t0) 76 0 for any time to, then
W(t) 76 O for all times, the solutions are linearly independent, and we can solve for
any initial conditions. The proof of the theorem is given at the end of this chapter.
An alternative proof is given later in the book using the Divergence Theorem from
calculus.

Theorem 2.3 (Liouville Formula). Let M(t) be a fundamental matvia: solution for a
linear system as given in equation (2.1) or equation (2.2), and let W(t) = det(M(t))

16 2. Linear Systezns

be the Wronskian. Then, and

iwo) = ti-(A(t)) W(t)

(2.3) W(t) = W(t0)exp (/ltr(A(s)) ds) ,

where exp(z) = e’ is the exponential function. In particular, if W(t0) 96 0 for any
time to, then W(t) 76 0 for all times t.

For a constant coe icient equation, if M(t) is a fundamental matrix solution,
the formula becomes

det(M(t)) = det(M(0)) etfim °.

Example 2.1. The Euler differential equation with nonconstant coef cients,

g; - 2:1;/+2y = 0.

can be used to give an example of the previous theorems. This second-order scalar
equation has solutions of the form t’, where 1' satis es

0=1'(r—l)—2'r+2='r2—3r+2=(r—1)(r—2).

Thus, there are two solutions y‘(t) = t and yz (t) = t2. This second-order differ-
ential equation corresponds to the rst-order system of differential equations

ii = $2,
:0. 2 _- t23 :01 + z2 2:2.

The preceding two scalar solutions correspond to the vector solutions

=<‘(¢) = (¢.1>T and x’(¢) = (¢’.2v)T,

which have Wronskian

W(t) = det 1‘ *22t = R '

On the other hand, the trace of the matrix of coef cients is 2/¢, so for t, to > 0,

W(t) = W(to) exp ds)

= tg exp (2 ln(t) — 2 ln(tQ))

= lg (e'“(‘2)/e'“(‘3)) = tg (£2/tg)

= n’.

Thus, the result from the Liouville formula (2.3) agrees with the calculation using
the solutions.

2.1. Fimdamental Set of Solutions 17

Matrix Exponential

For the constant coe icient case, equation (2.1), there is a general way of using
the exponential of a matrix to get a matrix solution that equals the identity when
t equals 0. This exponential is usually not easy to compute, but is a. very useful
conceptual solution and notation. Also, we shall use it to derive the form of the
solution when the eigenvalues are repeated.

For a scalar equation :i: = art, the solution is a:(t) = :00 e“‘ for an arbitrary
constant 1:0. For equation (2.1) with constant coef cients, we consider em as a
candidate for a matrix solution. This expression involves taking the exponential of
a matrix. What should this mean? The exponential is a. series in powers, so we write
A" for the 11°“ power of the matrix. Using the power series for the exponential, but
substituting a matrix for the variable, we de ne

6* - t" °° z"
3A‘:I+tA+—A2+...+1An+..-=£—An_
2! n! nl
1l=O

We will not worry about the convergence of this series of matrices, but in fact, it
does converge because of the factor of n! in the denominator. Differentiating with
respect to t term by term (which can be justi ed),

iaet A¢_ -Q+A+1! £A72+! A3+..(n.+—£1)!A"+...
=A I+tA+2! A2+---(+'n—lA1)"l “1+---

= A(eA‘).

Since em = I, eM is a fundamental matrix solution that equals the identity when
t equals 0. If v is any vector, then x(t) = emv is a solution, with x(0) = v.

We give a few special cases where it is possible to calculate em directly.

Example 2.2. Consider a. diagonal matrix,

A- (0a 0b).

<0 .>Then a" 0

and

M_ 1 0 at 0 l a2t2 0 i a."t" 0
e ‘(01)+(o bt)+2! 0 we +
+1.! 0 aw +

_ eat O

_ 0 eh‘ '

For any n xn diagonal matrix, the exponential of the matrix is also the diagonal
matrix whose entries are the exponential of the entriw on the diagonal. See Exercise
8 in this section.

18 2. Linear Systems

Example 2.3. Another simple example where the exponential can be calculated

explicitly is

B _ (O0J —0w).

Then,

IO =_
fa)
B2=__ _
=DEDJU(c\l 0E1@¢
Q)
T/-5 L_)L_)
B4: @‘GE ego “eJno

2n_ (_1)n"J2n 0
B_ 0 (__1)nw2n iv
' (__1)n+1w2n+1
Z-$[—?Z—\

BM“ : 0

(_1)nw2n+l 0 '

Writing out the series and combining the even and odd terms, we get

1 0 1 — 2152 0 1 4t‘ 0
em = <0
1) + i( all —w2t2> + I (W0 oft‘) +

1 (_1)nw2nt2'n 0_

+ (211,)! < 0 (-1)"t ":2" +"'

0 —wt 1 0 w3t3 1 0 —w5t5

l-(wt 0)+§l(—w3t3 0 )+§l(w5t5 0

1 0 (_1)1\+lw2n+1t2n+l

+ ((_1)nw2'n+lt2'n+1 0 ) +"'

= <cOsl)wt): c0sl()wt)) + (sinllut) _silIl)(wt))

_ (cos§wt) —sin(wt))
_ sin wt) cos(wt) '

Example 2.4.~ Let A =‘aI be a 2 X 2 diagonal matrix with equal entries and B
be as in the preceding example. Since AB = BA, Theorem 2.13 at the end of this
chapter shows that e(A+B)‘ = eA‘eB‘. (This formula is not true for matrices which
do not commute.) Therefore,

at —w t a. t 0 O —w t

exp wt at =exp O at exp wt O

_ e°' 0 cos(wt) — sin(wt)
_ 0 e“‘ sin(wt) cos(wt)

,,, cos(wt) — sin(wt)

=6 (sin(wt) cos(wt)

Example 2.5. Consider the matrix N~ = Then, N2 = 0 so N'° = 0 for

k 2 2. Therefore,

e"‘=I+N¢+0+---= ((1)

Exercises 2.1 19

Example 2.6. Consider the matrix A = aI+N = (8 (11) with N as the previous
example. Then,

eAt = ealteNt =eat

_ eat teat

_ 0 e°‘ '

It is only possible to calculate the exponential directly for a matrix in very

special “normal form", such as the examples just considered. Therefore, for linear

equations with constant coe icients, we give another method to nd a fundamental

set of solutions in the next section. For such a a fundamental set of solutions,

{x(3")(t)};-‘=1, let M(t) be the corresponding fundamental matrix solution, and

de ne ~

M(t) = M(t)M(0)"1.

By multiplying by the inverse of M(0), the initial condition when t = 0 becomes

the identity, M(0) = M(0)M(0)'1 = I. Because the inverse is on the right, this is
still a matrix solution, since

@=MmM@*

= lAM(¢)l MW)"
= A (t).

By uniqueness of solutions, IT/I(t) = em. Thus, it is possible to calculate em by

nding any fundamental set of solutions and constructing M(t) as in the preceding

discussion. '

Exercises 2.1

1. Show that the following functions are linearly independent:

1. 0 ,and 1
e‘ 1 , e2‘ 2 e" 0 .

0 1 1

2. Consider the system of equations

511 = $2-

iz = rqltl $1 — Pit) $2,

where q(t) and p(t) are continuous functions on all of R. Find an expression
for the Wronskian of a fundamental set of solutions.

3. Assume that M(t) is a. mdamental matrix solution of x = A(t)x and C is a
- constant matrix with det (C) 96 0. Show that M(t)C is a fundamental matrix
solution.

4. Assume that M(t) and N(t) are two fundamental matrix solutions of x =
A(t)x. Let C = M(0)“ N(0). Show that N(t) = M(t) C.

20 2. Linear Systems

5. (Hale and Kocak) Alternative proof of Theorem 2.13.
a. Prove that, if AB = BA, then Be“ = em B. Hint: Show that both
B em and e‘A B are matrix solutions of x = Ax and use uniqueness.
b. Prove Theorem 2.13 using part (a). Hint: Using part (a), verify that both

......_(0 0) .... .-(_, 0).e‘(A+B) and e‘A e‘B are matrix solutions of J‘: = (A + B) x.
01 0 0

a. Show that AB 76 BA.

b. Show that eA = and eB = <_11 Hint: Use Example 2.5

with t = 1 for eA and a similar calculation for eB .

c. Show that eAeB 76 eA+B. Hint: A +_B is of the form considered in

Example 2.3 ' -

7. Let A be an n X n matrix, and let P be a nonsingular n >< n matrix. Show that
ep -1 AP = P“eAP. Hi.nt: Wri.te out the power seri.es for ep - ‘AP.

8. Calculate the exponential em for the following diagonal matrices:

0'1 0 0 0.1 0 0
a. D= 0 0.2 0 ,
0 0.2 0
0 0 03 b. D= _ _ _ _

I: 2
O0 an

9. Calculate the exponential eN ‘ for the following matrices:

a.N=001,b.N= o10...oo
1...o0

,¢.N=___0_...00

;_/ 0...01
©©'@© 0...O0

Hint: (a) N3=0. (b) N4 =0. (c) N"=OifN isnxn.
10. Calculate the exponential eA‘ for the following matrices:

a1O 00

©@@@- 0. 1 00

a.A=Oa1,b.A= j O O 0. 0O
,c.A= _ __ __.

8D:Z? D@QQ OidQv—~ @\,@ @@@@@@ @@Q@|-@',_. @Q©l@d@,_,@ Pl—@*|©_r©@@ 000 a1

0 0 0 . .. 0 a

Hint: Write A = D + N where D and N are of the form of the preceding two
problems.

11. Assume that A is an n x n matrix such that AT = —A (antisymmetric). Prove

that eA is orthogonal, i.e., (eA)T eA = I.,,. Hint: Exponentiate the two sides
of the equation 0,, = A + AT where 0,, is the zero matrix. Use the fact that

ATA = (—A)A = A(—A) = AAT and e<A’> = (eA)T.

2. 2. Constant Coei cients 21

2.2. Constant Coefficients: Solutions and Phase Portraits

We motivate the form of the solutions of linear systems of equations by starting
with a linear scalar equation. The differential equation :i: = as: has a solution
x(t) = zone“ for arbitrary 20. If we combine two such scalar equations into a single
system, :i:; = a1a:1 and 1&2 = (12332, then it has a solution of the form a:1 (t) = c1e“1‘
and :n2(t) = c2e°", where cl and C2 are arbitrary constants. Combining these into
a vector solution,

x(t) = c1e““u1 + c2e°"u2,

where uj is the imit vector with a. 1 in the 3*“ place and 0 in the other entry. Notice
that this vector solution e°1‘u-‘I is always a scalar multiple of the vector uj, and it
moves along the line determined by uj .

With this as motivation, we try for a solution of the constant coef cient equa-

tion (2.1) that is of the form -

x(t) = e“v.

For it to be a solution, we need

Ae’\'v = % (e’\‘v) = ,\e*'v.

Since e'\‘ gé 0, this equality is equivalent to or
Av = /\v

(A — /\I)v = 0.

Here, I is the identity matria: with 1s down the diagonal and Os in the entries o ’ the
diagonal. Thus, x(t) = e'\‘v is a nonzero solution if and only if /\ is an eigenvalue
with eigenvector v. The way to nd such solutions is (i) to solve for eigenvalues /\_,-
as roots of the characteristic equation '

det(A — AI) = 0,

and then (ii) for each root, to solve for a corresponding eigenvector from the equa-
tion

(A. — /\_7'I)V = 0.

If there are n real distinct eigenvalues, A1, . . . , /\,,, with corresponding eigenvectors,
vl, . . . , v", then the eigenvectors are linearly independent and we get n solutions
that form a fundamental set of solutions. In this case, the general solution is of the
form

c1e""v1 + - - - + c,,e""’v".

For a two-dimensional linear system

ill _ 0. b Z131
ii; — C d I2 '

the characteristic equation is

0=det<a;'\ d /\)

=(a-,\)(d—).)—bc=,\2—(a+d)/\+(ad—bc)

=)\2—-r)\+A,

22 2. Linear Systems

where -r = a + d = tr(A) is the trace of the matrix and A = ad— be = det(A) is the
determinant. Therefore, the characteristic equation is especially easy to determine
in this case.

We proceed to solve some speci c examples for the fundamental set of solutions.
We also draw the representative solutions in the phase space.

Example 2.7 (Saddle). Consider

1i:=3131x'

This has characteristic equation A2 -— 2/\ — 8 = 0, with roots /\ = -2, 4.

For the eigenvalue /\1 = -2, A — ,\1I = A +'2I = . This can be row

reduced to 3), and so there is an eigenvector v1 /7 _o(.1oa_H1mw ). Thus, the rst
=

solution is given by

x1(t) = e'2' .

The components of this solution go to zero at the rate e‘2‘. The plot of the rst
component of this solution versus time, (t,a:}(t))T, is shown in Figure la. We call
this plot of one component of the solution versus t, a time plot of the solution.

2 111 21111

t t

F‘ Ii

1.6 ' 0.8

(a) (b)

211

t

1.6

(<1)

Figure 1. Time plot of solutions for Example 2.7 with a saddle: (a) (t, 2:1 (t))

for (r}(v).w§(@)> = (2-2). (b) (¢.=%(¢>> for (¢%(0).=%(@)) = (0-1.0-1). and

(c) (t,:cf(t)) for (=§(0),==§(0)) = (2,—1.995).

2. 2. Constant Coe cients 23

Similarly, the second eigenvalue Ag = 4 has an eigenvector v2 = The
second solution is given by

mt) = 6“ .

The components of the solution grow at a rate of e“. See Figure lb for a time plot
of this second solution, (t, a:§(t)).

The matrix solution from these two solutions is given by
e-2: 84:
_e-2: 84¢ -

The Wronskian of the system is

W(t) = det -<_e-e2_:,, Z4“: ) = 262‘.

Evaluating it at t equal to zero gives the determinant of the matrix with eigenvectors
as the columns:

w(o) = det(v1,v2) = det<_11 = 2 as o.

These two eigenvectors are independent, as must be the case from linear algebra,
since the two eigenvalues are di erent. Thus, the general solution is of the form

cle_2‘ (_11) +cge‘“ (11 ).

The plot of the variable a:1(t) as a fundtion of t for diiferent values of cl and cg is
given in Figure 1.

In addition to giving the time plot of the solution, it is often very instructive to
plot the solution in the (a:1,:cg)-plane as curves, without indicating the actual times,
but just the direction of motion by means of arrows. This plot of the solutions in
the x-space is called the phase portrait. When there are just two variables as in
this example, the x-space is also called the phase plane; in higher dimensions, it is
called the phase space.

Taking cl = 1 and cg = 0, we get the solution e‘2‘ (31), which moves along

the straight line of scalar multiples of the vector Notice that e'2‘ goes

to zero as t goes to in nity, but never equals zero, so the trajectory ed‘

approaches the origin as t goes to in nity, but it never actually reaches the origin.
This is evident in the time plot, Figure la. We can draw this trajectory in the
(:z1,a:g)-plane with an arrow to mark the direction to move along the curve. See
the straight line going toward the origin from the lower right in Figure 2. If we

set c1 = -1 and cg = 0, we get the solution e'2‘ <11), which also moves along a

straight Line from the upper left.

24 2. Linear Systems
V2

I2

I1

V1
Figure 2. Phase Portrait for Example 2.7 with a saddle

If we set cl = 0 and cg = 1, we get the solution e4‘ which also moves

along a straight line of scalar multiples of its eigenvector, but it grows in size as t
increases: This solution goes to in nity as t goes to in nity, and goes to the origin
as t goes to minus in nity. This trajectory is along the half-line in the upper right

of Figure 2. If we set <c1 = 0 and c2 = -1, we get the solution e“ which

moves away from the origin along the half-line in the lower left of Figure 2.
For a solution that contains components along both eigenvectors, cl 94 0 and

C2 #01 _’*(»11)~»@"<1.)=¢**lc~_@‘"l<-1)~»1 (1)l~

the term inside the square bracket approaches cg as t goes to in nity, so the

trajectory has the line generated by the eigenvector as an asymptote as t

goes to in nity. Similarly, the trajectory has the line generated by the eigenvector

(J1) as an asymptote as t goes to minus in nity. For example, if both c1 and cg

are positive, the trajectory is like the one moving on the hyperbola at the right of

Figu_re 2. For .

n) = 1.997518 (t) + 0.0025 x2(t),

with x3(0) = (2, —1.995), the plot of (t,_a:‘1’(t)) is given in Figure lc.

If cl < 0 and cg > O, the trajectory is like the one at the top of Figure 2.

The origin for such a linear system with one positive real eigenvalue and one
negative real eigenvalue is called a saddle. Sometimes we abuse terminology by
calling the linear system a saddle.

2.2. Constant Coe icients 25

Table 1 surmnarizes the procedure for drawing the phase portrait near a saddle
linear system in two dimensions.

Phase portrait for a saddle linear system

(1) Draw all the trajectories that move along straight lines in or out
of the origin by nding the eigenvectors. Mark each of these half-
lines with the direction that the solution is moving as t increases.

(2) In each of the fo1u' regions between the straight line solutions,
draw in representative solutions, which are linear combinations of
the two primary solutions.

When using a computer program, such as Maple, Mathematica, or
Matlab, to draw the phase portrait, certain steps are helpful. Assume
that /\1 < 0 < /\g are the eigenvalues, with corresponding eigenvectors
v1 and v2.

1. Take two initial conditions that are small multiples of :l:_v1 and
follow them forward in time (e.g., :b0.01v1).

2. Take two other initial conditions :tcv2 that are large scalar mul-
tiples of ivz, so the points are about on the edge of the region
displayed and follow the solutions forward in time. (Or, take two .
initial conditions that are small multiples of ivz and follow them
backward in time.)

3. Take at least four other initial conditions that are slightly off the
scalar multiples of :l:v2 (e.g., ztcvz zt 0.1 v1).

‘Table 1

‘I

Assume we have a system in 1R2 with two distinct negative real eigenvalues,
O > A1 > A2, with corresponding eigenvectors vl and v2. Then, the general
solution is of the form

c1e’\"vl + c2e’\="v2.

The solutions with either" (i) cl 96 0 and Cg = 0 or (ii) c1 = 0 and C2 74 0 move
toward the origins along straight lines. If both c1, C2 76 0, then the solution can be
written as

e’\“(c1v1 + cge(’\’—"“)‘v2).

Because /\2 < A1, A2 — /\1 < 0 and the second term in the sum goes to zero as t
goes to in nity. This shows that any solution with cl 7é 0 approaches the origin
asymptotic to the line generated by the eigenvector v1 (i.e., asymptotic to the line
generated by the eigenvector corresponding the eigenvalue that is less negative). As
t goes to minus in nity, the term involving e’\" goes to zero faster and the solution
becomes more parallel to the line generated by the vector v2. Figure 3 is a sketch
of the phase portrait, which exhibits the four trajectories that move along straight
lines toward the origins, as well as other solutions that have both c1,c¢ 96 O. The
time plots of the solutions all go to zero at an exponential rate; however, the second
solution goes at a faster rate.

26 2. Linear Systems

The origin for a system of this type (or the linear system itself) with all eigen-
values real, negative, and distinct is called a stable node.

‘$2 v2

‘,1

-‘B1

Figure 3. Phase portrait for Example 2.9 with a stable node

Remark 2.8. The word node means a point at which subsidiary parts originate.
It is used in both biology for the point on the stem where a leaf is inserted, and
in electrical circuits for the points where the di erent branches of the circuit come
together. In the preceding phase plane, the solutions branch out from the origin;
hence, the name. » ._-

The adjective stable is used because all the solutions tend to the origin as t
goes to infinity.

Example 2.9 (Stable Node). The system

it = (_34 —T121 ) x

is an example having a stable node. The matrix has eigenvalues A1 = —5 and

/\2 = *10 with corresponding eigenvectors v1 = and v2 = The general

solution is _.. (12) We_ 1

with phase portrait as shown in Figure 3. The Wronskian of the solutions at t equal
zero is given by

W(0)=det 21 31 =5¢0;

thus, the two eigenvectors are independent, so the two solutions are independent.

Table 2 summarizes the proced1u"e for drawing the phase portrait near a stable
node with distinct eigenvalues in two dimensions.

2. 2. Constant Coefficients 27

Phase portrait for a stable node

We assume that the eigenvalues are distinct, 0 > M > A2.

(1) Draw all the trajectories that move along straight lines toward the
origin. Mark each of these half-lines with the direction that the
solution is moving as t increases. Mark each of the trajectories
approaching the origin at the faster rate (corresponding to the
more negative eigenvalue) with a double arrow.

(2) In each of the four regions between the straight line solutions,
draw in representative solutions, which are linear combinations
of the two primary solutions. Make sure that, as the trajectories
approach the origin (as t goes to in nity), they are tangent to the
line generated by the eigenvector corresponding to the eigenvalue
which is less negative. Make sure that, as the trajectories go olf
toward in nity (as t goes to minus in nity), they become parallel
to the eigenvector corresponding to the eigenvalue which is more
negative.

When using a computer program to draw the phase portrait, certain
steps are helpful. Assume 0 > /\1 > Ag are the eigenvalues with corre-
sponding eigenvectors v1 and v2.

1. Take two initial conditions ic1 v1 that are large multiples of iv‘
so the points are about on the edge of the region displayed and
follow the solutions forward in time.

2. Take two other initial conditions Il.‘.C2V2 that are large scalar mul-
tiples of iv’, ItC2V2, so the points are about on the edge of the
region displayed and follow the solutions forward in time.

3. Ta.ke at least one initial cdndition in each of the fo1u' regions
separated by iv‘ ivz that are on the edge of the region displayed
and follow the solutions forward in time.

Thble 2

Example 2.10 (Unstable Node). If both eigenvalues of a system on IR2 are
positive and distinct, then the origin for the linear system is called an unstable
node. The phase portrait is similar to the preceding example, with the direction of
the motion reversed. See Figure 4.

Drawing the phase portrait for an imstable node is similar to a stable node
with obvious changes: the arrows are reversed and the trajectory converges to the
origin as t goes to minus in nity.

Example.2.11‘ (Zero Eigenvalue). The differential equation

1': = -—11 --33 x

28 2. Linea: Systems
(E2 V2
V1
$1

l

Figure 4. Phase portrait for Example 2.10 with an unstable node

has ei.genvalues 0 and -4. The vector _31 1. s an e.igenvector for 0 and 11 1. 5
an eigenvector for —4. Since e°‘ = 1, the general solution is I

x(t) = c1 + c2e"“ .

Notice that all pointshthat are multiples of are xed points. (These are the

initial conditions obtained by taking C2 = 0.) Since e—‘“ goes to zero as t goes to
in nity, all other solutions tend to this line of xed points. See Figure 5.

vl

Figure 5. Phase portrait for Example 2.11 with a zero eigenvalue
The origin is always a xed point for a linear equation, 5: = A.x,_ since A0 = 0.
The only way that there are other xed points is for 0 to be an eigenvalue; if